Lemon Lover's Cookie (holiday baking #4)

This is my second non-chocolate recipe... and I'm not a big fan of lemon (in my cookies) but I have to say this one literally almost melts in your mouth.  It could be due to the amount of butter in it :)  - I have definitely been channeling my southerner with all this baking!! :)

• Cookie:
• 3/4 cup butter, softened
• 3 tablespoons sugar
• 2 teaspoons lemon juice
• 1 cup all-purpose flour
• 1/2 cup cornstarch
• 1 teaspoon grated lemon peel
• 
  Lemon Frosting:
• 1/4 cup butter, softened
• 1 cup confectioners' sugar
• 2 teaspoons lemon juice
• 1 teaspoon grated lemon peel
• 
  optional garnish:
• Additional grated lemon peel, optional
• 
  
• In a small bowl, cream butter and sugar until light and fluffy, about 5 minutes. Beat in lemon juice. Combine the flour, cornstarch and lemon peel; gradually add to creamed mixture and mix well.
• 
  
• Shape into a 1-1/2-in. roll; wrap in plastic wrap. Refrigerate for 1 hour or until firm. Unwrap and cut into 1/4-in. slices. Place 2 in. apart on silpat lined baking sheets.
• 
  
• Bake at 350° for 10-12 minutes or until edges are golden brown. Gently remove to wire racks to cool completely.
• 
  
• Frosting: In a small bowl, beat butter until fluffy. Add the confectioners’ sugar, lemon juice and peel; beat until smooth. Spread over cooled cookies; sprinkle with additional lemon peel if desired. Let stand until set. Store in an airtight container.